Maximum-likelihood analysis of axial displacement in fluorescence phase-shifting interferometry

Fluorescence phase-shifting interferometry (FPSI) is an optical technique that coherently combines the phase-shifted 4π steradian emission wavefronts of a single fluorescent emitter to obtain multiple interferograms from which the emitter axial displacement can be retrieved with high precision. Here, we study the axial displacement sensitivity in 4-step FPSI within the framework of maximum-likelihood (ML) phase estimation. Using Monte-Carlo simulations, we show that regardless of the method used to preprocess the measured interferograms, the variance of the ML estimate of the axial displacement approaches the Cramér-Rao lower bound and is closely limited from above by the variance of the classical 4-step phase shifting estimator. The difference between these lower and upper bounds depends on the interferogram visibility and signal-to-noise-ratio (SNR), with a percentage change of up to 29% that yields an absolute change of a few sub-nanometers to some nanometers for SNRs larger than ~5. Our results suggest that for these levels of SNR, the use of the computationally simpler classical 4-step phase shifting estimation can be adequate to accurately determine axial displacements in FPSI, as we also experimentally verified. FPSI interferograms with lower SNR but good visibility can benefit from the use of the ML estimator, provided that the spatiotemporal phase stability of the FPSI system is high.